Utility has a standard deviation of 40 percent, and the return on the Commodity has a
standard deviation of 30 percent. However, she does not know the exact covariance in the
returns of the two stocks. Emmy would like to plot the variance of the portfolio for each
of three cases—covariance of 0.12, 0, and –0.12—in order to understand how the
variance of such a portfolio would react. Do the calculation for all three cases (0.12, 0 and
–0.12), assuming an equal proportion of each stock in the portfolio.
LO 5
Solution:
12
2
1
2
2
2
2
2
1
2
1
2
2
)
(
σ
σ
σ
x
x
x
x
R
Var
port
asset
+
+
=
Case
1, σ
12
= 0.12:
(0.5)
2
× (0.4)
2
+ (0.5)
2
× (0.3)
2
+ 2 × (0.5) × (0.5) × (0.12) = 0.1225
Case
2, ρ = 0.0:
(0.5)
2
× (0.4)
2
+ (0.5)
2
× (0.3)
2
+ 2 × (0.5) × (0.5) × (0.0) = 0.0625
Case
3, σ
12
= -0.12:
(0.5)
2
× (0.4)
2
+ (0.5)
2
× (0.3)
2
+ 2 × (0.5) × (0.5) × (-0.12) = 0.0025